Finding the limit points of $N_{\infty}$ if its a union of a set of natural numbers and $\infty$

Question

Kindly see what i did, and let me know if there is anything else I need to add or modify in my proof. I am hoping that I am good, but not sure.

Any help will be greatly appreciated.

Answer 1

You're quite right: all points of $\Bbb N$ are isolated as $\{0\} = \{x: x < 1\}=[0,1)$ and $\{n\}=(n-1,n+1)$ for $n \ge 1$. While $\infty$ has basic neighbourhoods of the form $(n,\infty]=\{x: x > n\}$ for some $n \in \Bbb N$. This is an infinite set (or just remark it contains $n+1 \neq \infty$ etc.) so $\infty$ is a limit point (the only one!) of $\Bbb N_\infty$.

So your solution is correct, but can be written more concisely. We don't need to consider subspaces (as you do), but only the order topology on $\Bbb N_\infty$ (in the order as described); we want to know its limit points, so its non-isolated points. Mentioning $\Bbb R$ is a red herring.